Understanding the coupling between convective clouds and the general circulation, as well as addressing the gray zone problem in convective parameterization, requires insight into the genesis and maintenance of spatial patterns in cumulus cloud populations. In this study, a simple toy model for recreating populations of interacting convective objects as distributed over a two‐dimensional Eulerian grid is formulated to this purpose. Key elements at the foundation of the model include i) a fully discrete formulation for capturing discrete behavior in convective properties at small population sample sizes, ii) object age‐dependence for representing life‐cycle effects, and iii) a prognostic number budget allowing for object interactions and co‐existence of multiple species. A primary goal is to optimize the computational efficiency of this system. To this purpose the object birth rate is represented stochastically through a spatially aware Bernoulli process. The same binomial stochastic operator is applied to horizontal advection of objects, conserving discreteness in object number. The applicability to atmospheric convection as well as behavior implied by the formulation is assessed. Various simple applications of the BiOMi model (Binomial Objects on Microgrids) are explored, suggesting that important convective behavior can be captured at low computational cost. This includes i) subsampling effects and associated powerlaw scaling in the convective gray zone, ii) stochastic predator‐prey behavior, iii) the downscale turbulent energy cascade, and iv) simple forms of spatial organization and convective memory. Consequences and opportunities for convective parameterization in next‐generation weather and climate models are discussed., Plain Language Summary: Convective clouds play a crucial role in Earth's climate. The way they interact with the atmospheric circulation is not well understood, and is associated with long‐standing problems in weather forecasting and climate prediction. Recent research has suggested that the spatial structure of these cloud fields is a key factor in this problem, and that improving our understanding of such convective cloud patterns is crucial for making progress. This study explores a new model framework for generating such cloud patterns, consisting of populations of convective objects on small grids. The objects are born in a random way, complete a life cycle, and can freely move around on the grid. They can also interact and form larger clusters, obeying certain rules of interaction. The way the objects behave and move around features some key innovations compared to previous ecosystem models of this kind. These are introduced to optimize the performance and reduce run time on a computer. Various experiments are conducted to explore the new model, illustrating that well‐known behavior of convective populations is reproduced. These tests also highlight opportunities created for improving convection in weather and climate models., Key Points: A scale‐aware stochastic number generator based on a Bernoulli process is applied to model object births and advection on Eulerian grids. Discreteness in object number is conserved, while an age dimension is included to represent object life cycle effects. Population subsampling effects in the convective gray zone are reproduced, while simple applications capture well‐known convective behavior., U.S. Department of Energy (DOE) http://dx.doi.org/10.13039/100000015